ManyPrimates pilot mixed modeling

library(tidyverse)
library(knitr)
library(sjPlot)
library(gridExtra)
library(lme4)
library(emmeans)
library(car) # for vif
library(bbmle) # for AICtab
library(broom) # for glance
theme_set(ggthemes::theme_few())

Summary

Mixed modeling with all relevant variables predicting accuracy

From the preregistration, the mixed model was specified thusly:

correct ~ delay * age + 
          task_experience + cup_distance + board_size + trial +
          (1 + delay + trial | site/subject/block/hiding_location ) + 
          (1 + task_experience + cup_distance + board_size + trial + delay | species)

In the dataframe, subject_site = subject, and norm_age should be used for age.

Model as pre-registered has too many random effects

Error: number of observations (=6246) < number of random effects (=10608) for term (1 + delay + trial | hiding_location:(block:(subject_site:site))); the random-effects parameters are probably unidentifiable

Pruning random effects in the following order (from preregistration):

• Remove correlations between random effects
• Remove random slopes (in the following order)
  • species
  • hiding_location
  • block
  • subject

Model only converges once we take out hiding_location. After doing so, the other random effects (correlation, site, species) can be put back in.

The model below converges. Model output is saved in 06_mp_model_v2.rds

correct ~ delay * norm_age + 
          task_experience + cup_distance + board_size + trial + 
          (1 + delay + trial | site/subject_site) + 
          (1 + task_experience + cup_distance + board_size + trial + delay | species)

Reduced model

After pruning random effects with little variability and removing board_size, which covaried with cup_distance, the reduced model has the following structure. It is saved in 06_mp_3_model3_v2.rds

correct ~ delay * norm_age + 
          task_experience + cup_distance + trial + 
          (1 + delay | site/subject_site) + 
          (1 + delay | species)

---

Data prep

Data import

mp_data <- read.csv("../data/merged_data/01_manyprimates_pilot_merged_data_v2.csv")

Prepare code for pre-registered mixed modeling

• center cup_distance, board_size and trial
• filter out spider monkey. Only one data point so far, therefore this is not worth including to explode the number of random effects

model.data <- mp_data %>%
  filter(species != "black_faced_spider_monkey") %>%
  mutate_at(vars(cup_distance, board_size, trial), funs(scale(.)[, 1])) %>%
  mutate(hiding_location = factor(hiding_location),
         delay = fct_relevel(delay, "short"))

Model 1

The model takes a while to run. Run next line to load model output from previous run with structure below.

# mm.1 <- readRDS("06_mp_model.rds")
mm.1 <- readRDS("06_mp_model_v2.rds")

mm.1 <- glmer(correct ~ delay * norm_age +
               task_experience + cup_distance + board_size + trial +
               (1 + delay + trial | site/subject_site/block) +
               (1 + task_experience + cup_distance + board_size + trial + delay | species)
             , data = model.data
             , family = binomial
             , control = glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e5))
             )

saveRDS(mm.1, "06_mp_model_v2.rds")

Some diagnostics

• examining Cholesky decomposition

theta <- getME(mm.1, "theta")
diag.element <- getME(mm.1, "lower") == 0
any(theta[diag.element] < 1e-5)

Model summary

Confirm model structure

# mm.1@call
formula(mm.1)

correct ~ delay * norm_age + task_experience + cup_distance + 
    board_size + trial + (1 + delay + trial | site/subject_site) + 
    (1 + task_experience + cup_distance + board_size + trial + 
        delay | species)

glance(mm.1) %>% kable(digits = 2)

Random effects

VarCorr(mm.1) %>% print(formatter = fmt, digits = 3) # comp = c("Variance", "Std.Dev.")

 Groups            Name               Std.Dev. Corr                         
 subject_site:site (Intercept)        0.862                                 
                   delaylong          0.646    -0.91                        
                   delaymedium        0.528    -0.85  0.99                  
                   trial              0.093    -0.31  0.61  0.71            
 site              (Intercept)        0.917                                 
                   delaylong          0.512    -1.00                        
                   delaymedium        0.627    -1.00  1.00                  
                   trial              0.084     1.00 -1.00 -1.00            
 species           (Intercept)        0.835                                 
                   task_experienceyes 0.173    -1.00                        
                   cup_distance       0.012     1.00 -1.00                  
                   board_size         0.232    -1.00  1.00 -1.00            
                   trial              0.021    -1.00  1.00 -1.00  1.00      
                   delaylong          0.494    -1.00  1.00 -1.00  1.00  1.00
                   delaymedium        0.436    -1.00  1.00 -1.00  1.00  1.00
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
  1.00

Fixed effects

CIs

mm.1.ci <- confint(mm.1, method = "Wald") %>% # bootstrap these later
  as.data.frame %>%
  rownames_to_column %>%
  filter(complete.cases(.)) %>%
  rename(LL = `2.5 %`, UL = `97.5 %`) %>%
  mutate(OR_LL = exp(LL), OR_UL = exp(UL))

coef(summary(mm.1)) %>%
  as.data.frame %>%
  rownames_to_column() %>%
  mutate(OR = exp(Estimate)) %>%
  left_join(mm.1.ci, by = "rowname") %>%
  select(rowname, OR, OR_LL, OR_UL, Estimate, LL, UL, everything()) %>%
  kable(digits = 3)

corr <- cov2cor(vcov(mm.1)) %>% as.matrix %>% round(2)
corr[upper.tri(corr, diag = T)] <- ""
colnames(corr) <- 1:10
rownames(corr) <- str_c(1:10, " ", rownames(corr))

corr %>% as.data.frame %>% select(-10) %>% rownames_to_column

Pairwise contrasts for delay

based on estimated marginal means

Note. This wasn’t in the preregistration.

emmeans(mm.1, pairwise ~ delay, type = "response")$contrasts

 contrast       odds.ratio         SE  df z.ratio p.value
 short / long    3.5160746 0.92828519 Inf   4.762  <.0001
 short / medium  2.7103806 0.75716209 Inf   3.569  0.0010
 long / medium   0.7708541 0.07335892 Inf  -2.735  0.0172

Results are averaged over the levels of: task_experience 
P value adjustment: tukey method for comparing a family of 3 estimates 
Tests are performed on the log odds ratio scale 

Model 1 plots

Fixed effects

plot_model(mm.1, title = "Fixed Effects", order.terms = c(7, 4, 3:1, 9:8, 5, 6),
           width = .3, show.values = T, value.size = 2.5, value.offset = .3) +
  geom_hline(yintercept = 1, lty = 2) +
  ylim(0, 3)

Random effects

ranef.plots <- plot_model(mm.1, type = "re", sort.est = "(Intercept)")

Subject/Site

ranef.plots[[1]]

Site

Species

---

Pruning the model

• remove trial random slopes within species as the estimates in the previous models were essentially 0

correct ~ delay * norm_age + 
          task_experience + cup_distance + board_size + trial +
          (1 + delay + trial | site/subject_site ) +         
          (1 + task_experience + cup_distance + board_size + delay | species)

Check colinearity in the previous model

vif(col.mm1)

                    GVIF Df GVIF^(1/(2*Df))
delay           1.007876  2        1.001963
norm_age        1.072757  1        1.035740
task_experience 1.056383  1        1.027805
cup_distance    1.323990  1        1.150648
board_size      1.296698  1        1.138726
trial           1.001067  1        1.000533

No signs of high colinaearity.

Check levels of random effects

Check how many different levels there are within each random effect

overview$summary

$`delay_within_species (factor)`

 3 
11 

$`task_experience_within_species (factor)`

1 2 
9 2 

$`board_size_within_species (covariate)`

1 2 4 
5 5 1 

$`cup_distance_within_species (covariate)`

1 2 4 
6 4 1 

$`trial_within_species (covariate)`

36 
11 

This suggests that, within species, random slopes for task_experience does not make much sense as most species have only 1 level. Same is true for cup_distance and board_size. Indeed, the model summary and random effects plot for species confirm that there is little variability in these estimates (they’re close to zero). Therefore they are removed.

correct ~ delay * norm_age + 
          task_experience + cup_distance + board_size + trial +
          (1 + delay + trial | site/subject_site ) +         
          (1 + delay | species)

Model 2

The model takes a while to run. Run next line to load model output from previous run with structure below.

mm.2 <- readRDS("06_2_mp_model2_v2.rds")
# mm.2.ci <- readRDS("06_2_mp_model2_ci_v2.rds")

mm.2 <- glmer(correct ~ delay * norm_age +
              task_experience + cup_distance + board_size + trial +
              (1 + trial + delay | site/subject_site) +
              (1 + delay | species)
              , data = model.data
              , family = binomial
              , control = glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e5))
              )

saveRDS(mm.2, "06_2_mp_model2_v2.rds")

Model 3

Remove trial from the random slopes for subject/site as it’s near zero both in mm.1 and even more so in mm.2

VarCorr(mm.2) %>% print(comp = c("Variance", "Std.Dev."), formatter = fmt, digits = 3)

 Groups            Name        Variance Std.Dev. Corr             
 subject_site:site (Intercept) 0.74     0.86                      
                   trial       0.009    0.093    -0.29            
                   delaylong   0.417    0.646    -0.91  0.60      
                   delaymedium 0.279    0.528    -0.85  0.69  0.99
 site              (Intercept) 1.052    1.026                     
                   trial       0.008    0.089     0.88            
                   delaylong   0.349    0.591    -1.00 -0.88      
                   delaymedium 0.471    0.686    -1.00 -0.89  1.00
 species           (Intercept) 0.576    0.759                     
                   delaylong   0.246    0.496    -1.00            
                   delaymedium 0.183    0.428    -1.00  1.00      

The model takes a while to run. Run next line to load model output from previous run with structure below.

mm.3 <- glmer(correct ~ delay * norm_age +
              task_experience + cup_distance + board_size  + trial +
              (1 + delay | site/subject_site) +
              (1 + delay | species)
              , data = model.data
              , family = binomial
              , control = glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e5))
              )

saveRDS(mm.3, "06_3_mp_model3_v2.rds")

Model summary

Confirm model structure

formula(mm.3)

correct ~ delay * norm_age + task_experience + cup_distance + 
    board_size + trial + (1 + delay | site/subject_site) + (1 + 
    delay | species)

sigma	logLik	AIC	BIC	deviance	df.residual
1	-3391.26	6838.52	7027.23	6389.41	6218

LRT

drop1(mm.3, test = 'Chisq')

Single term deletions

Model:
correct ~ delay * norm_age + task_experience + cup_distance + 
    trial + (1 + delay | site/subject_site) + (1 + delay | species)
                Df    AIC     LRT  Pr(Chi)    
<none>             6836.2                     
task_experience  1 6834.4  0.1483   0.7001    
cup_distance     1 6854.5 20.3087 6.59e-06 ***
trial            1 6835.0  0.7541   0.3852    
delay:norm_age   2 6832.6  0.3836   0.8255    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Random effects

VarCorr(mm.3) %>% print(comp = c("Variance", "Std.Dev."), formatter = fmt, digits = 3)

Fixed effects

CIs

# this is not currently run
source("boot_glmm.r")

mm.3.ci <- boot.glmm.pred(model.res = mm.3, excl.warnings = F, nboots = 1000,
                         para = F, resol = 100, level = 0.95, use = NULL,
                         circ.var.name = NULL, circ.var = NULL, use.u = F,
                         n.cores = c("all-1", "all"), save.path = NULL)

saveRDS(mm.3.ci, "06_3_mp_model3_ci_v2.rds")

# mm.3.ci <- confint(mm.3, method = "Wald") %>% # bootstrap these later
#   as.data.frame %>%
#   rownames_to_column %>%
#   filter(complete.cases(.)) %>%
#   rename(LL = `2.5 %`, UL = `97.5 %`) %>%
#   mutate(OR_LL = exp(LL), OR_UL = exp(UL))

	rowname	OR	Estimate	X2.5.	X97.5.	Std. Error	z value	Pr(>|z|)
(Intercept)	(Intercept)	5.805	1.759	0.978	2.541	0.448	3.925	0.000
delaylong	delaylong	0.240	-1.426	-1.943	-0.939	0.280	-5.092	0.000
delaymedium	delaymedium	0.313	-1.161	-1.663	-0.665	0.281	-4.130	0.000
norm_age	norm_age	0.992	-0.008	-0.212	0.211	0.109	-0.078	0.938
task_experienceyes	task_experienceyes	1.016	0.015	-0.282	0.350	0.173	0.089	0.929
cup_distance	cup_distance	1.462	0.380	0.169	0.613	0.103	3.677	0.000
board_size	board_size	1.472	0.386	0.198	0.593	0.094	4.097	0.000
trial	trial	1.024	0.023	-0.037	0.086	0.030	0.774	0.439
delaylong:norm_age	delaylong:norm_age	1.052	0.051	-0.162	0.249	0.106	0.479	0.632
delaymedium:norm_age	delaymedium:norm_age	1.093	0.089	-0.117	0.286	0.101	0.883	0.377

Pairwise contrasts for delay

based on estimated marginal means

(emm <- emmeans(mm.3, specs = pairwise ~ delay)$contrasts)

 contrast         estimate        SE  df z.ratio p.value
 short - long    1.4257372 0.2800475 Inf   5.091  <.0001
 short - medium  1.1604769 0.2810297 Inf   4.129  0.0001
 long - medium  -0.2652602 0.0865917 Inf  -3.063  0.0062

Results are averaged over the levels of: task_experience 
Results are given on the log odds ratio (not the response) scale. 
P value adjustment: tukey method for comparing a family of 3 estimates 

emm

 contrast         estimate        SE  df z.ratio p.value
 short - long    1.4257372 0.2800475 Inf   5.091  <.0001
 short - medium  1.1604769 0.2810297 Inf   4.129  0.0001
 long - medium  -0.2652602 0.0865917 Inf  -3.063  0.0062

Results are averaged over the levels of: task_experience 
Results are given on the log odds ratio (not the response) scale. 
P value adjustment: tukey method for comparing a family of 3 estimates 

(emm.or <- emmeans(mm.3, pairwise ~ delay, type = "response")$contrasts)

Error in is.data.frame(data) : object 'model.data' not found
Error in ref_grid(object, ...) : 
  Perhaps a 'data' or 'params' argument is needed

confint(emm.or)

 contrast       odds.ratio         SE  df asymp.LCL asymp.UCL
 short / long    4.1609240 1.16525642 Inf  2.158448 8.0211731
 short / medium  3.1914550 0.89689360 Inf  1.651737 6.1664689
 long / medium   0.7670063 0.06641638 Inf  0.626125 0.9395867

Results are averaged over the levels of: task_experience 
Confidence level used: 0.95 
Conf-level adjustment: tukey method for comparing a family of 3 estimates 
Intervals are back-transformed from the log odds ratio scale 

Model 3 plots

Fixed effects

ggsave("../graphs/05_forestplot.png", fe.3, width = 3, height = 2.5, scale = 2)

Random effects

ranef.plots2 <- plot_model(mm.3, type = "re", sort.est = "(Intercept)")

Subject/Site

ranef.plots2[[1]]

Site

ranef.plots2[[2]]

Species

Model 4

• further remove subject/site random effects to look at species differences (from preregistration)

mm.4 <- glmer(correct ~ delay * norm_age +
                task_experience + cup_distance + trial +
                (1 + delay | species)
              , data = model.data
              , family = binomial
              , control = glmerControl(optimizer = "bobyqa", optCtrl = list(maxfun = 2e5))
        )

Random effects

phylo <- read_csv("../data/species_data.csv") %>% 
  select(species, species_formatted, phylo)

plot.data <- get_model_data(mm.4, type = "re", show.values = T) %>%
  left_join(phylo, by = c("term" = "species")) %>%
  rename(species = species_formatted) %>%
  mutate(
    facet = fct_rev(facet),
    facet = fct_recode(facet, "Intercept" = "species (Intercept)",
                       "Delay (short vs. long)" = "delaylong", 
                       "Delay (short vs. medium)" = "delaymedium")
    )

Column `term`/`species` joining factor and character vector, coercing into character vector

ggsave("../graphs/05_forestplot_fe_re.png", arrangeGrob(fe.3, re.4, layout_matrix = mat), width = 8, height = 3, scale = 1.8)

ggsave("../graphs/Fig3.tiff", arrangeGrob(fe.3, re.4, layout_matrix = mat), width = 8, height = 3, scale = 1.8, type = "cairo", compression = "lzw")

Model comparison

We’re looking for the lowest AIC(c) as the model with the ‘best fit’ with a reasonable number of parameters. (Too many are penalized by AIC as one way to address overfitting.)

Indeed, the reduced model seems to do a better job of striking that balance between fitting the data with fewer parameters.

AICctab(mm.1, mm.2, mm.3, mm.4, logLik = T, weights = T)

     dLogLik dAICc df weight
mm.3 109.9     0.0 28 0.9982
mm.2 111.6    12.7 36 0.0018
mm.1 112.9    54.9 58 <0.001
mm.4   0.0   193.6 15 <0.001

anova(mm.1, mm.2, mm.3, mm.4)

Data: model.data
Models:
mm.4: correct ~ delay * norm_age + task_experience + cup_distance + 
mm.4:     trial + (1 + delay | species)
mm.3: correct ~ delay * norm_age + task_experience + cup_distance + 
mm.3:     board_size + trial + (1 + delay | site/subject_site) + (1 + 
mm.3:     delay | species)
mm.2: correct ~ delay * norm_age + task_experience + cup_distance + 
mm.2:     board_size + trial + (1 + trial + delay | site/subject_site) + 
mm.2:     (1 + delay | species)
mm.1: correct ~ delay * norm_age + task_experience + cup_distance + 
mm.1:     board_size + trial + (1 + delay + trial | site/subject_site) + 
mm.1:     (1 + task_experience + cup_distance + board_size + trial + 
mm.1:         delay | species)
     Df    AIC    BIC  logLik deviance    Chisq Chi Df Pr(>Chisq)    
mm.4 15 7032.3 7133.4 -3501.1   7002.3                               
mm.3 28 6838.5 7027.2 -3391.3   6782.5 219.7490     13     <2e-16 ***
mm.2 36 6851.0 7093.6 -3389.5   6779.0   3.5119      8     0.8983    
mm.1 58 6892.6 7283.5 -3388.3   6776.6   2.4403     22     1.0000    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Difference in regression coefficients

Difference

coef2 - coef1

   delaylong  delaymedium cup_distance 
 -0.16840056  -0.16341754  -0.06131301 

Difference in odds ratios

exp(coef2) - exp(coef1)

   delaylong  delaymedium cup_distance 
 -0.04407273  -0.05561161  -0.09242183